This and (REF ) are exactly the conditions of Theorem 1 in Chapter II.8 of [1]} which we stated in the Appendix as Theorem REF  for the convenience of the reader. The theorem states that the sequence \(\widetilde{p}_n(x):=\sum _{k=0}^{m_n}c_kL_k(x)\)  with \(c_k=\int _0^{\infty }p(x)L_k(x)e^{-x}dx\)  converges locally uniformly, i.e., uniformly on compact sets, to \(p\) . Let
\(\mathcal {C}_n:=\int _0^{\infty }e^{-x}\widetilde{p}_n(x)^2dx=\sum _{k=0}^{m_n}c_k^2\) 
